Definition 3.2 (Local differential privacy [1]}) 

Let \(\mathcal {X}\)  and \(\mathcal {Y}\)  denote the spaces of original and noisy data, respectively. A randomizing mechanism \(\mathcal {R}\)  provides \(\epsilon \) -local differential privacy (LDP) if, for all \(x,\,x^{\prime }\,\in \,\mathcal {X}\) , and all \(y\,\in \,\mathcal {Y}\) , we have
\(\mathbb {P}[\mathcal {R}(x)=y] \le e^{\epsilon }\,\mathbb {P}\left(\mathcal {R}(x^{\prime })=y\right)\) 

